A) \[(1-\cos {{a}^{3}})\]
B) \[3(1-\cos {{a}^{3}})\]
C) \[-\frac{1}{3}(1-\cos {{a}^{3}})\]
D) \[\frac{1}{3}(1-\cos {{a}^{3}})\]
Correct Answer: D
Solution :
\[I=\int_{0}^{a}{{{x}^{2}}\sin {{x}^{3}}dx}\]; Put \[{{x}^{3}}=t\Rightarrow {{x}^{2}}dx=\frac{dt}{3}\] \[\therefore \,\,\,I=\frac{1}{3}\int_{0}^{{{a}^{3}}}{\sin t\,dt}=-\frac{1}{3}[\cos t]_{0}^{{{a}^{3}}}=-\frac{1}{3}[\cos {{a}^{3}}-1]\] \[=\frac{1}{3}[1-\cos {{a}^{3}}]\].
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